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Theorem srglz 17373
Description: The zero of a semiring is a left-absorbing element. (Contributed by AV, 23-Aug-2019.)
Hypotheses
Ref Expression
srgz.b  |-  B  =  ( Base `  R
)
srgz.t  |-  .x.  =  ( .r `  R )
srgz.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
srglz  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )

Proof of Theorem srglz
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 srgz.b . . . . . . . 8  |-  B  =  ( Base `  R
)
2 eqid 2454 . . . . . . . 8  |-  (mulGrp `  R )  =  (mulGrp `  R )
3 eqid 2454 . . . . . . . 8  |-  ( +g  `  R )  =  ( +g  `  R )
4 srgz.t . . . . . . . 8  |-  .x.  =  ( .r `  R )
5 srgz.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
61, 2, 3, 4, 5issrg 17354 . . . . . . 7  |-  ( R  e. SRing 
<->  ( R  e. CMnd  /\  (mulGrp `  R )  e. 
Mnd  /\  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  (
( x  .x.  (
y ( +g  `  R
) z ) )  =  ( ( x 
.x.  y ) ( +g  `  R ) ( x  .x.  z
) )  /\  (
( x ( +g  `  R ) y ) 
.x.  z )  =  ( ( x  .x.  z ) ( +g  `  R ) ( y 
.x.  z ) ) )  /\  ( (  .0.  .x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) ) )
76simp3bi 1011 . . . . . 6  |-  ( R  e. SRing  ->  A. x  e.  B  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
87r19.21bi 2823 . . . . 5  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  ( A. y  e.  B  A. z  e.  B  ( ( x  .x.  ( y ( +g  `  R ) z ) )  =  ( ( x  .x.  y ) ( +g  `  R
) ( x  .x.  z ) )  /\  ( ( x ( +g  `  R ) y )  .x.  z
)  =  ( ( x  .x.  z ) ( +g  `  R
) ( y  .x.  z ) ) )  /\  ( (  .0. 
.x.  x )  =  .0.  /\  ( x 
.x.  .0.  )  =  .0.  ) ) )
98simprd 461 . . . 4  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (
(  .0.  .x.  x
)  =  .0.  /\  ( x  .x.  .0.  )  =  .0.  ) )
109simpld 457 . . 3  |-  ( ( R  e. SRing  /\  x  e.  B )  ->  (  .0.  .x.  x )  =  .0.  )
1110ralrimiva 2868 . 2  |-  ( R  e. SRing  ->  A. x  e.  B  (  .0.  .x.  x )  =  .0.  )
12 oveq2 6278 . . . 4  |-  ( x  =  X  ->  (  .0.  .x.  x )  =  (  .0.  .x.  X
) )
1312eqeq1d 2456 . . 3  |-  ( x  =  X  ->  (
(  .0.  .x.  x
)  =  .0.  <->  (  .0.  .x. 
X )  =  .0.  ) )
1413rspcv 3203 . 2  |-  ( X  e.  B  ->  ( A. x  e.  B  (  .0.  .x.  x )  =  .0.  ->  (  .0.  .x. 
X )  =  .0.  ) )
1511, 14mpan9 467 1  |-  ( ( R  e. SRing  /\  X  e.  B )  ->  (  .0.  .x.  X )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   0gc0g 14929   Mndcmnd 16118  CMndccmn 16997  mulGrpcmgp 17336  SRingcsrg 17352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273  df-srg 17353
This theorem is referenced by:  srgmulgass  17377  srgrmhm  17382
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